 Hi friends, I welcome you all to another session on quadratic equation actually it's you can't really say that it's a session on quadratic equation we are because we are going to discuss today something called infinitly extending nested radicals nested radicals okay what is a nested radical so if you see here it is under root 6 plus under root 6 plus under root 6 plus and it is going on till infinity now this kind of radical is called infinitly extending nested radical general form of the same general form of the same will be n plus under root n plus under root n plus one and so forth right so basically we are going to try and find the value of this yeah and interestingly the value comes out to be in such cases in many cases actually the value comes out to be a rational number that's surprising is it so it looks like it's a irrational number but the value comes out to be rational let's check how and how quadratic equation knowledge is going to help us find the value in fact the famous mathematician Ramanujam also invested a lot of time in infinitely extending nested radicals and he has given lots of theorems around it so but here we will be keeping it simple and let's try to find out the value so I am going to assign a variable x and I'm going to call this as also the value which is expected to be found out I am assigning that to a variable x okay now the best part of a nested variable is whether you because it's extending till infinity so the value of the entire nested radical and the value within it you know these two right so hence if you if you really call it x you can expect or you can you know rewrite it as 6 plus x itself why because you see this thing itself you defined as x and since it is it is happening only because it is extending till infinity so hence this item as a whole you defined as x so hence I can write x equals under root 6 plus x and then that's the simple you know how to deal with such equations so you square both sides you'll get x square is equal to 6 plus x and this becomes x square minus x minus 6 equals 0 so hence there are two roots again so x square minus 3x plus 2x minus 6 equals 0 so this is xx minus 3 plus 2x minus 3 equals 0 right so hence what is this this is nothing but x minus 3 times x plus 2 equals 0 right so finally what will the value of x be x is 3 or x is minus 2 so now we are going to reject this value x equals to minus 2 because root square root of anything is has been defined as positive so whenever we say square root of 4 we mean 2 and never minus 2 isn't it so don't get confused so whenever you see this root 4 you should write 2 and never minus 2 then you will say how come because minus 2 square is also 4 but that comes from here so if you write x square is equal to 4 then you can write x equals to plus minus 2 when you're solving this equation but somewhere if root 4 is given only value is 2 and never write this as minus 2 so here also we'll reject this value so x is 3 okay so hence root of 6 plus root of 6 plus root of 6 plus infinite such radicals is equal to 3 isn't it interesting that something which was looking or appearing to be a irrational number ends up being a rational number yep so actually if you see this is a concept of limit so hence as here as it goes towards infinity the value tends to the value 3 and hence this is a limiting value of this particular nested radical right you can generalize this actually right so how to generalize so if you see if you have this n plus under root n plus under root n plus so on and so forth till infinity the value and hence yeah you have to be careful that here n is greater than 1 so if n is greater than 1 I'm sorry n is greater than 0 not 1 so n has to be a positive quantity so if n is a positive quantity n is positive quantity then what will happen you this value will be nothing but half times 1 plus under root 1 plus 4 n okay you can check right now what was n n was 6 so in n is 6 so this value will be nothing but half into 1 plus under root 1 plus 24 4 times 6 24 right so which is nothing but half times 1 plus 5 and hence it is 3 so it matches isn't it so this is how so hence this is a generalized form so please remember next time when you come across such kind of a thing you please remember this result is it fine so this is what is called nested radical more on nested radical can be found out in a video session under radicals section